Why tensor opeator?

In books about angular momentum, they introduce the so call tensor operator to deal with angular momentum, but why's that and what does it look like? In the cover page of J.J.Sakurai's textbook, there is a block matrices, is that any relation to tensor operator?

I'm not really sure what tensor operator means in your context, my guess is that take two operators in two hilbert spaces, say
[tex]a \textrm{ for } \mathcal{H}[/tex]
and
[tex]b \textrm{ for } \mathcal{H'}[/tex]

For example, we can take a one particle hilbert space, take it's tensor product so that we have a two particle hilbert space. Then we can get operators that acts on this hilbert space by specifiying it's action on the first particle and the second particle. Of course, more general operator may not be tensor products of operators.

" Notice that the effect of multiplying the unit vector by the scalar is to change the magnitude from unity to something else, but to leave the direction unchanged. Suppose we wished to alter both
the magnitude and the direction of a given vector. Multiplication by a scalar is no longer sufficient. Forming the cross product with another vector is also not sufficient, unless we wish to limit the change in direction to right angles. We must find and use another kind of mathematical ‘entity.’ "